Euclid’s Proof of the Infinitude of Primes: Distorted, Clarified, Made Obsolete, and Confirmed in Modern Mathematics

REINHARD SIEGMUND-SCHULTZE

n this article1 I reflect on the recurrent theme of Michael Hardy and Catherine Woodgold (henceforth modernizing historical mathematical proofs, vocabulary, H&W) entitled ‘‘Prime Simplicity’’ appeared in this journal II and symbolism, and the extent to which this modern- in 2009. H&W discuss Euclid’s proof mainly from the ization serves to clarify, is able to preserve, or is bound to point of view of its distortion by modern authors who distort the original meaning. My example is also a quite often claim that it had the form of a reductio ad recurrent one: Euclid’s proof of the infinitude of primes absurdum.3 in book IX, theorem 20 of his Elements. Elementary My article, on the other hand, aims at a systematic pre- number theory is a very appropriate field for discussing sentation and logical analysis of Euclid’s proof (as it is such general historiographic questions on a nontechnical preserved in the critical editions by Euclid scholars) and, level.2 Indeed, a widely quoted and stimulating article by above all, at a more detailed discussion of its interpretation

1I dedicate this article to Walter Purkert (Bonn), the coordinator and very active force behind the excellent Felix Hausdorff edition soon to be completed, on the occasion of his 70th birthday in 2014. 2Mesˇ trovic´ (2012) contains many details on modern proofs of the infinitude of primes but is less interested in Euclid’s original theorem and its modern variants. Most of his material goes far beyond the simple fact of the infinitude of primes that is the focus of the present paper. 3In the following, the notions reductio ad absurdum, proof by contradiction, and indirect proof are used interchangeably for assuming the logical opposite of what one wants to prove and then reaching, by legitimate logical steps, a contradiction that requires the assumption of the original claim.

2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 87 DOI 10.1007/s00283-014-9506-9 and rewriting by a few select modern mathematicians, that the Ancients ‘‘proved the irrationality of the square root foremost among them Dirichlet, G. H. Hardy, and Hilbert.4 of two’’ cannot be unqualifiedly maintained either. More As I will show, Euclid’s proof was constructive. For all recently, discussions were raised about the so-called the emphasis put on properties and axiomatic structure as ‘‘geometric algebra,’’ that is, the claim (for instance defen- displayed in Euclid’s Elements, constructions—finding, in ded by B. L. van der Waerden and meanwhile refuted by finitely many legitimate steps (i.e., those allowed by axioms historians) that some of Euclid’s geometric theorems have and theorems), a mathematical object with certain prop- to be interpreted as translations of historically preexisting erties—is the backbone of Greek mathematics.5 More algebraic problems and equations.6 specifically, I will argue that Euclid’s proof is ‘‘weakly Historians know, of course, that absolute historical constructive,’’ and it is partly this weak form of construc- ‘‘accuracy’’ cannot be attained in modern presentations and tiveness—the fact that Euclid does not provide an effective that they have to refer to modern vocabulary to make the method (formula) to calculate new primes from given very process of creation understandable.7 In discussing ones—which provided modern mathematicians with the Euclid’s Elements, the problem starts with the question as to wrong impression that Euclid’s proof was indirect. what extent it is legitimate to replace the purely verbal I agree with H&W that the insinuation of an indirect proof original formulation throughout the book by some other, in Euclid IX, 20 became the ‘‘prevailing doctrine’’ when modernized one that uses symbols. In the special case of modern mathematicians wanted to relate in a positive way to the Pythagorean-Euclidean number theory, Euclid’s dress- Euclid’s geometry. However, modern presentations of ing up of the theorems in geometric clothing, without Euclid’s proof, such as the ones by Hardy in 1938 and 1940, genuinely geometric content, adds to the problem. are not genuine indirect proofs of the infinitude of primes. In the case of Euclid’s Elements, one is best served They are quite close in spirit and content to Euclid’s original relying on Bernard Vitrac’s careful French edition (1990– proof. 2001) with very detailed commentary,8 which is still based (as is the most famous English edition by Thomas Heath What Does ‘‘Projecting Modern Mathematics from 1908 and 1925) on the Greek text, edited by the into the Past’’ Mean, and How Much Historical Danish historian J. L. Heiberg in 1883–1885.9 Vitrac says in Accuracy Can or Should One Reach in Modern his edition that—because of a scarcity of original sources— Texts? each modern specialist of Greek number theory provides a Mathematicians and historians projecting modern mathe- different interpretation (Vitrac 1994: 288). This should warn matics into the past can miss essential points of the us, the nonspecialists, to restrict our discussion basically to intentions and results of the historical creators of mathe- the modern reception of what the specialists agree is a matics. To mention a few: the modern notion of functional reliable text of Euclid’s theorem IX, 20 and other theorems dependence and its graphic representation originated related to it. (basically) in the Scientific Revolution of the 17th century, Vitrac’s and Heath’s commentaries (and others) do not connected to the need to extend the realm of mathematical contribute many answers to my questions: to what extent is objects to some ‘‘mechanical’’ curves (higher algebraic Euclid’s theorem IX, 20 constructive or its proof indirect, curves treated by Descartes) and to formulate the laws of what kind of infinity does it claim, and how has the physics (Newton). Projecting that modern notion uncriti- theorem been interpreted by modern mathematicians. cally into the past does not, however, help us much to Nevertheless I will try to use their and other Euclid scholars’ understand Apollonius’s sophisticated theory of conics. expertise to provide nuancing counterbalance against some Even Euclid’s Elements, which, because of their axiomatic formulations that necessarily have to be somewhat simpli- structure, have a more familiar appeal to modern mathe- fying so as to be short and understandable. maticians than Apollonius’s works, cause heated My first question is admittedly speculative and cannot be methodological-historical debates in their modern inter- answered in this article (although an answer would shed pretations. For example, the deep theory of geometric light on our problems): Why did Euclid prove the ‘‘infini- proportions in book V of the Elements caused R. Lipschitz, tude of primes’’ in the first place? in his discussion with R. Dedekind in the 1870s, to claim its The proof comes in book IX of the Elements, as prop- logical equivalence with the modern definition of real osition 20, almost at the end of the ‘‘number-theoretic’’ numbers. The latter, being the inventor of the Dedekind (‘‘Pythagorean’’) books VII through IX, whereas the deci- cut, of course denied that claim, and most historians agree sive instruments and propositions, particularly the with him (Nikolic´ 1974). One consequence is that the claim Euclidean algorithm and propositions close (though not

4The focus of the article is not, however, on the reception of Euclid’s work during the more than 2000 years since its creation, which would require special linguistic (including Arabic!), philologic, and philosophical qualifications. Although H&W refer, mostly critically and summarily, to an impressive number of 147 publications, I will quote many fewer modern mathematicians. However, I will refer to a few additional ones (Hilbert, Weil). In addition I will use historical literature, above all B. Vitrac’s edition of the Elements. 5More on that later, in connection with an analysis of Euclid’s proof. 6Concerns about this notion were already articulated in the late 1960s by A. Szabo´ , indicated in Knorr (1975), and formulated polemically by Unguru (1975). 7We will speak frequently, for instance, about ‘‘sets of primes’’ below, without, however, assuming modern set-theoretical operations and notions coming with it. 8Of course the French language might constitute a barrier to the reception of Vitrac’s edition in our modern, largely English-speaking world. We will in the following refer to volume 2 of this edition, containing books V–IX of the Elements, as Vitrac (1994). 9Cf. June Barrow-Green (2006) for an overview of earlier English editions of the Elements.

88 THE MATHEMATICAL INTELLIGENCER fully equivalent) to the fundamental theorem of arithmetic, substantially by contradiction. Various forms of MIP are presented in the beginning and in the middle of book (Modern Infinitude Proofs) sometimes are substantially VII. The algorithm and the propositions are frequently used indirect proofs (as in the cases of Dirichlet [1863] and Hardy in Euclid’s number-theoretic books, but his theorem on the [1908] to be discussed later) and sometimes claim to be ‘‘infinitude’’ of primes is not used at all. The remaining indirect although they follow closely the Euclidean original propositions of book IX, 21 through 36, are mainly about (as in the case of another proof by Hardy of 1938 and the properties of odd and even numbers.10 Thus theorem 1940). IX, 20 appears isolated. Did Euclid see it as the demon- To be sure, indirect proofs were not foreign to the stration of a kind of ideal, platonic existence of infinitely Greeks (they rather invented them) and many Greek many primes, something like the last theorem of the Ele- philosophers/scientists (among them mathematically ments, proposition XIII, 18, in which he proves that there highly educated individuals such as Plato and Eudoxus, exist no more than 5 platonic polyhedra?11 That might who stood behind Euclid’s Elements with their spirit and/ explain why a ‘‘weakly constructive’’ proof sufficed for him: or results) speculated about infinity.14 Andre´ Weil, who he apparently needed the additional primes only in a was deeply interested in the history of number theory, general ‘‘philosophical’’ sense, not to prove other pointed to book X, def. 3 of the Elements to show that theorems. Euclid was not averse in principle to talking about infinity. At the same time, Weil uses that example to What Did Euclid Really Claim and Prove and What Is caution readers not to expect too much impact of the Its Modern Counterpart? philosophical positions of Greek mathematicians on their Euclid does not really claim or prove the infinitude of work: primes. What he says and proves in IX, 20 is: The views of Greek philosophers about the infinite may ECL (Euclid’s claim): be of great interest as such; but are we really to believe ‘‘Prime numbers are more than any assigned multitude that they had great influence on the work of Greek of prime numbers.’’ mathematicians? Because of them, we are told, Euclid It may appear pedantic and trivial to stress,12 but it is had to refrain from saying that there are infinitely many historically a relevant fact that it requires a very simple, yet primes, and had to express fact that differently. How is it indirect argument to derive from this statement the fol- then that, a few pages later, he stated that ‘‘there exist lowing modern claim: infinitely many lines’’ incommensurable with a given MCL (Modern claim of the infinitude of primes, in the one?15 Cantorian sense of actual infinity): Thus one may never fully know whether Euclid added in ‘‘There exist infinitely many primes.’’ his mind such an indirect proof, leading him from ECL To derive the MCL we make the (wrong) assumption (which he proved) to MCL. But this question does not that the finite ‘‘assigned multitude’’ is ‘‘complete’’ (contains affect at all the problem of whether his proof, as we all natural numbers that fall under the definition of a prime have inherited it, was (logically) indirect, that is, by number). That is, the (finite) ‘‘assigned multitude’’ must contradiction. contain all primes up to a maximum, which is not stipulated What we know for sure, however, is Euclid’s proof of in Euclid’s original proof. This explains why most modern ECL and that it was done by a (weak) construction that can presentations of Euclid’s proof use uninterrupted sequen- be considered as performing the following assignment: ces of primes (beginning with 2) and their products ECO (Euclid’s construction): (primorials) as starting points. Although the infinitude fol- ‘‘Given a finite number of primes, find at least one lows easily from contradiction, one could also argue that additional prime.’’ this conclusion of infinitude can be drawn ‘‘constructively’’ Constructions occur in almost all theorems in Euclid’s in the sense of a continued counting of a potentially infinite Elements, at least as auxiliary methods. Constructions are set of prime numbers.13 usually also applied in proofs of theorems that do not ask Modern misinterpretations of ECL in the sense of MCL for constructions or that do not talk about the existence of lead to the insinuation that Euclid’s proof was basically or mathematical objects, but only about properties. The

10This was also observed by David E. Joyce in his online version of Heath’s English edition, which is quoted later. Vitrac (1994: 86), calls these last theorems ‘‘enigmatic,’’ meaning that their connection to the preceding theorems of the arithmetic books is problematic anyway. 11This comparison seems also relevant with respect to our central question of constructiveness: the construction of the five platonic polyhedra in propositions 13 through 17 of book XIII requires incomparably more effort than the ‘‘negative’’ result in proposition 18 that there are no more than the five regular polyhedra. Like Euclid’s theorem on the infinitude of primes, his last theorem in book XIII, 18 obviously remains without application in the Elements, because it comes last. Vitrac (1994: 273) cautions against jumping too easily to the assumption that Euclid was a Platonist. 12We do not go quite so far though as putting ‘‘proof of the infinitude of primes’’ into quotation marks in the title. 13The resulting infinitude would, of course, not necessarily be complete. Cf. for instance Graham/Knuth/Patashnik (1989: 108) where such an infinitude is recursively constructed with the help of the Euclidean algorithm and the auxiliary notion of relatively prime ‘‘Euclid numbers.’’ Vitrac (1994: 445) stresses that this infinity is fully compatible with Aristotle’s notion of potential infinity. 14However, in mathematics itself they usually treated infinity indirectly (as for instance in Eudoxus’ famous notion of proportion in book V of the Elements) and preferred the safety of the finitely many steps in mathematical constructions. Of course this is related to the fact that the Greeks did not possess modern analysis. 15Weil (1978: 230). Knorr (1975: 233) refers to the remark in Euclid as an ‘‘anomaly,’’ because it is not a definition but seems to promise a theorem for which one has, however, to ‘‘wait in vain.’’

2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 89 majority of theorems in the Elements are of this kind.16 The [smaller] number’’) by ‘‘product’’ of numbers. Both notions most famous example is probably the ‘‘theorem of Pytha- occur in Euclid; they are differently defined (Vitrac 1994: goras’’ in book I, theorem 47. Some propositions, however, 258), the first based on properties, the second on construc- are explicitly formulated as construction assignments, and tion, but they are logically equivalent. the latter—combined with their solution that then fol- Euclid’s proof is then based on two simple lemmas (L1 and lows—have to be understood as ‘‘construction theorems.’’ L2), and on one not-too-deep theorem (T). Whereas theorem The best known of this kind in the ‘‘number-theoretic’’ T is proposition VII, 31 in Euclid’s Elements, the two lemmas books VII–IX is the so-called ‘‘Euclidean algorithm,’’ do not explicitly appear in the Elements, but they follow easily namely proposition VII, 2: from the notion of the product of natural numbers. ‘‘Given two numbers not prime to one another, to find their greatest common measure.’’17 L1: The product of any two numbers (none of them the Expressed in modern words, in Euclid’s number theory, unit) is bigger than either of them. constructions are usually performed by the four funda- L2: A product of n numbers (none of them the unit) + 1 mental operations of arithmetic, in particular addition and is not divisible by any of these numbers. multiplication, which in the Elements appear in geometric T: Any product of numbers (none of them the unit) is clothing.18 But also in the number-theoretic books one divisible by a prime number. finds some tendency to hide constructions in favor of properties. Thus ECO is not formulated separately in the L2 refers to a legitimately constructed number P + 1, Elements but it appears as the proof method in proposition which is the main idea of the proof, T continues IX, 20 with its claim ECL. I will argue in the following also the construction based on the possibility of repeated division. that the use of the notion of the least common multiple in ECO instead of the more constructive notion of a product Euclid, Elements, book IX, proposition 20: seems to indicate this predilection for logical structure and ECL: ‘‘Prime numbers are more than any assigned contributes to the underestimation of the constructive multitude of prime numbers.’’ 19 character of Euclid’s theorem of the infinitude of primes. Proof (ECO):

Let A, B, and C be the assigned prime numbers. I say that there are more prime numbers than A, B, and C.

Euclid’s Proof of ECL in IX, 20 Based on ECO Take the least number DE measured by A, B, and C. Add It is necessary to recall Euclid’s proof in some detail as a point the unit DF to DE. of reference to understand which elements or steps of it have Then EF is either prime or not. been changed or deleted in modern proofs. Some analysis of First, let it be prime. Then the prime numbers A, B, the structure and methods of Euclid’s proof is also needed. C,andEF have been found, which are more than A, B,andC. The following short summary of proof methods is (for the Next, let EF not be prime. Therefore it is measured by convenience of modern readers) based on one simplification some prime number [VII, 31].20 compared to the original formulation. As will become clear Let it be measured by the prime number G. from the text that follows, the summary replaces Euclid’s I say that G is not the same with any of the numbers A, notion of ‘‘composite’’ number (‘‘which is measured by some B, and C.

16There is an old historical discussion among historians of Greek mathematics (Wilbur Knorr, David Fowler, etc.) that continues today about the relation between constructive and deductive principles in the mathematics of the time and the influence of philosophy on that relationship. 17This algorithm was certainly mathematically deeper than ECO and ECL, and Donald Knuth called it the ‘‘granddaddy of all algorithms’’ (Knuth 1981: 318). The Euclidean algorithm is not needed for Euclid’s proof in IX, 20, although finding the greatest common divisor is closely related to finding the least common multiple LCM, which is assumed in IX, 20. The reason is that the LCM is trivial for prime numbers. However, as remarked earlier, Graham/Knuth/Patashnik (1989: 108) use the Euclidean algorithm to construct an infinitude of primes recursively. 18The essentially arithmetic content of the ‘‘number-theoretic books VII–IX’’ in the Elements is stressed by Knorr (1976). Vitrac (1994, 277) emphasizes differences in detail between the constructions of book I in the Elements and the constructions in the number-theoretic books. 19David E. Joyce uses Heath’s edition from 1908 for his online edition at the website of Clark University, from which we have quoted here: http://aleph0.clarku.edu/*djoyce/java/elements/elements.html. Figures have been added in Heath/Joyce and in some other editions for explanation. Heath/Joyce follow closely the Greek version in J. L. Heiberg’s edition of 1883–1885, replacing the Greek letters in the same alphabetic order. Cf. page 271 of the bilingual online edition by Richard Fitzpatrick at http://farside.ph.utexas.edu/euclid/elements.pdf. Although it does not include genuine geometric content, the figure is useful, because the text speaks of the least common multiple DE, and one would expect the unit to be called EF, not DF. The picture explains that. I thank June Barrow-Green for this observation. 20Vitrac 1994, 444, refers here to VII, 32 instead, which is a simple conclusion of VII, 31, but does not contain the construction of the divisor. Explicit reference to previous theorems (as explanation) is usually not contained in Euclid’s text and is not added in Heiberg’s Greek edition.

90 THE MATHEMATICAL INTELLIGENCER If possible, let it be so. Now A, B, and C measure H&W make much of the fact that Euclid does not DE, therefore G also measures DE. But it also measures expressly say that the proof uses the assumption that S is EF. Therefore G, being a number, measures the remainder, complete. However Euclid (and many other mathemati- the unit DF, which is absurd. cians with him, even today) does not always fully explain Therefore G is not the same with any one of the num- what he does and intends. We conclude, as a matter of bers A, B, and C. And by hypothesis, it is prime. Therefore course, that in Euclid the three numbers A, B, C stand for the prime numbers A, B, C, and G have been found, which an arbitrary finite set of primes, the cardinality of which are more than the assigned multitude of A, B, and C. we would today describe by the indeterminate number n. Therefore, prime numbers are more than any assigned We take for granted that Euclid saw that the new prime multitude of prime numbers. he had ‘‘constructed’’ was B P + 1. So why not assume Q. E. D. that he had thoughts about the completeness of S as well? Paraphrasing of the Proof: We do, however, have a compelling and very simple Given any finite set S of primes, one considers (con- argument to discard the hypothesis that Euclid assumed S 22 structs) their product P [Euclid uses for some not fully to be complete. This argument lies, of course, in the clear reason21 the LCM instead of the product; R. S.-S.] proof itself. In it Euclid concludes with the help of L1 that and adds the unit 1. If P + 1 is a prime, one has found P + 1 lies outside S. But he then considers as one of two an additional prime, which means a prime that is not in possible cases the one that P + 1 could be prime. But this the original set S (L1). If P + 1 is not a prime, it is is a conclusion that Euclid never would have drawn under divided by a prime (T). This latter prime cannot, how- the assumption that S was complete. To perform a reductio ever, be one of the primes in the original finite set either, ad absurdum requires, of course, concluding correctly because in this case it could not divide P + 1 (L2). from a hypothesis that one wants to disprove. Indeed, Di- Therefore we have also in this case found an additional richlet in his (indirect) proof (see later), which assumes the prime that does not lie in the original set S of prime completeness of S, discards immediately the possibility of numbers assumed. P + 1 being prime. This is a paraphrasing of Euclid’s proof (or of what has Note that Euclid’s proof, although not globally assuming come down to us in handwritten copies and in various the completeness of primes, nevertheless uses locally Arabic, Latin, and modern translations of the Greek ‘‘indirect arguments’’ as steps in the proof. And this use original), which follows closely Heiberg’s Greek edition occurs even on two levels, one open and one concealed. In of Euclid’s text, although partly using modern vocabulary the middle of the proof, the assumption of P + 1 being (set). divisible by a prime in S is refuted with the help of lemma It is remarkable that the use of lemma L1 is not expressly L2. And on a more hidden level: The proof of theorem T mentioned or argued within Euclid’s proof. It is simpler ‘‘Any product of numbers (none of them the unit) is than L2 but in its content closely related to it. It shows that divisible by a prime number’’ uses an indirect argument, P + 1 (regardless of whether P is the LCM or the product of which in Heath’s English edition appears within the proof all given primes) is bigger than any prime in S and therefore of VII, 31 as: cannot be one of these primes. However, L1 is often Thus, if the investigation be continued in this way, some emphasized in modern presentations of the theorem, for prime number will be found which will measure the instance in the one by Dirichlet to be mentioned in the text number before it, which will also measure A. For, if it is that follows. L1 is necessary for Euclid’s proof to show that not found, an infinite series of numbers will measure the P + 1 lies outside S. number A, each of which is less than the other: which is impossible in numbers. Vitrac in his French edition of the Elements emphasizes that Is Euclid’s Proof One by Contradiction (in we have an infinity argument here that is reminiscent of the Addition to Being One by Construction)? ‘‘descent infinite which has been made famous by Fermat.’’ An indirect proof of the infinitude of primes has to be (Vitrac 1994: 341). To be sure, this ‘‘hidden’’ indirect based on the (false) assumption that the given finite set of argument is of no immediate concern when deciding primes is ‘‘complete.’’ Now the question arises as to whe- whether Euclid’s proof in IX, 20 is ‘‘indirect,’’ because ther Euclid’s proof could at the same time—in addition to theorem T (= VII, 31) is accepted in the proof as a being weakly constructive—be interpreted as an indirect mathematical fact. However, the use of T nevertheless one, by assuming that S comprises all primes and by dis- contributes to the ‘‘feeling’’ that we have to do with indirect proving this assumption. In other words, could one arguments, because theorem T is obviously partly respon- conclude MCL immediately from the original proof ECO? sible for the ‘‘weak’’ constructiveness of IX, 20. This is even After all, it seems at first glance that S could well have been more true of the unconcealed indirect argument mentioned assumed by Euclid to be both finite and complete. Thus the before. Because it seems difficult to rewrite these two latter assumption would have been disproved by ECO and indirect arguments in the form of a direct conclusion Euclid would have produced a contradiction. (which would imply that the indirect argument is merely

21The reason may be again the emphasis of properties over constructions. 22H&W mention this as well as an additional argument (p. 46).

2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 91 superficial),23 we are left with the result that the proof IX, our hypothesis, and the latter has therefore to be dis- 20 contains locally genuinely indirect arguments, but carded.’’ Hardy does not provide a definition of a prime globally it is not indirect because it does not assume the number in his 1908 book either, which would be completeness of the given finite set of primes. important to know for reconstructing his argument. He apparently concluded along these lines: The constructed Insinuating Contradiction and (Almost) Ignoring number,whichIwillagaincallP + 1, is not divisible + Construction: In Particular in G. H. Hardy by ‘‘any prime’’ in S (L2). P 1 can therefore—because of the completeness of S (assumption A) and theorem H&W note that Euclid’s proof is misinterpreted by ‘‘no less T—not be composite and must be prime. But then it a number theorist’’ than G. H. Hardy (1877–1947) in his follows from the completeness of S and from L2 that Course of Pure Mathematics (1908). Because they do not P + 1 cannot be divisible by itself and thus cannot be quote Hardy in detail and because this fascinating example prime. This is a contradiction and it follows ‘‘non-A’’ serves my further argument, I present here the full passage (symbolically A), which is the original claim one wants from the original of 1908, which is repeated in subsequent to prove. editions up to the sixth of 1933:24 Symbolically Hardy’s indirect modern proof of the Euclid’s proof is as follows. If there are only a finite infinitude of primes MIP-H1908 can thus be written in the number of primes let them be l, 2, 3, 5, 7, 11, … N. Con- following way, although he does not refer to the second sider the number l + (l. 2. 3. 5. 7. 11 … N). This number is step of the argument in his verbal formulation: evidently not divisible by any of 2, 3, 5, … N, since the L2þT L2 remainder when it is divided by any of these numbers is 1. ðH 1908Þ½A ƒƒƒ!ðP þ 1Þprime^ ½A ƒƒƒ!ðP þ 1Þ It is therefore not divisible by any prime save l, :prime!:AÞ and is therefore itself prime [my emphasis, R. S.-S.], which is contrary to our hypothesis (Hardy 1908: Compared to Euclid, Hardy—in addition to assuming the 122/123). completeness of S (assumption A)—exchanges the steps of Hardy, unlike Euclid, uses primorials in his proof, that is, the proof, and he does not use lemma L1 at all. Instead of the products pn# of the first n primes. We will discuss this (similar to a hypothetical indirect proof in Euclid) first con- change (maybe even distortion) further below. In addition, sidering the case of P + 1 being prime and recognizing it as a Hardy assumes these n primes to form a complete set of contradiction to A because of L1, Hardy considers first P + 1 primes (‘‘there are only’’). Surprisingly, Hardy considers the being composite and discards it as a contradiction to A number 1 as a prime. But what is really striking is the claim because of L2. By exchanging the steps of proof, he is left that P + 1 ‘‘is therefore itself prime.’’ An inexperienced with the conclusion that P + 1 is prime under the assump- student can easily jump to the conclusion that P + 1 must tion of A, which is then finally recognized as a contradiction. indeed be prime whatever prime number N one starts with. One could argue that Hardy liked to play the logical This is, of course, not the case, as the famous and sim- game to the end and to the extreme, namely to conclude plest counterexample, p6# + 1 = 2 9 3 9 5 9 7 9 11 9 the absurd statement that P + 1 is ‘‘prime’’ under the 13 + 1 = 30,031 = 59 9 509, shows. Of course, we should (wrong) assumption of A. He seems, however, less inter- not assume that the great English number theorist was not ested in the constructive conclusion of Euclid’s proof, aware of the counterexample. To the contrary, he might which considers the divisibility of P + 1 as a composite well have been ‘‘too aware’’ of it, thus committing a number. Thus, Hardy distorts Euclid’s proof ECO in several ‘‘pedagogical error.’’ Indeed, the conclusion P + 1 ‘‘is respects, assuming the completeness of S, using primorials, therefore itself prime’’ might be convincing for the begin- and changing the steps in Euclid’s arguments. And one ner for two reasons: it is ‘‘confirmed’’ by the first five realizes that the order of steps and the wording matter 25 ‘‘Euclid numbers’’ up to p5# + 1 = 2311, and it is sup- when one analyses the various forms of the presentation of ported by the intuitive, if misled, feeling that there cannot Euclid’s proof. be a divisor of P + 1 between N and P + 1. However, the Compare now Hardy’s proof with the one by Dirichlet in conclusion ‘‘is therefore itself prime’’ will be considered the posthumous edition of his Number Theory of 1863.26 ‘‘absurd’’ by the mathematically educated and thus com- Starting with the maximum prime p in the finite set S (which pletes for him the reductio ad absurdum. Because one can is assumed to be complete) and (like Hardy after him) correctly deduce any statement from a wrong one, the using primorials, Dirichlet continues: conclusion that P + 1 must be prime is one that should not Each number greater than p must be composite and be too surprising for the educated mathematician. hence divisible by at least one of these primes. But it is Hardy does not bother to extend the last sentence of very easy to construct a number greater than p and not his proof with a remark such as ‘‘…which is contrary to divisible by any of the primes: just construct the product

23Note that any constructive proof can be rewritten as an indirect proof by assuming the opposite. But it should be superficial to acknowledge the latter as a genuine indirect proof. 24I have checked the sixth edition of 1933 and can confirm that Hardy changes the passage for the first time in the seventh edition of 1938, which he calls ‘‘revised and re-set.’’ See below. 25This is historically clearly a misnomer. ‘‘Euclid number’’ as used in Graham/Knuth/Patashnik (1989) is, however, somewhat closer to Euclid. 26Dirichlet 1863 and 1999, pp. 15–16, edited by R. Dedekind 1863 and translated into English by J. Stillwell in 1999. This is mentioned without discussion in Hardy (2013), which contains some additional remarks and slight corrections to H&W (2009).

92 THE MATHEMATICAL INTELLIGENCER of the primes from 2 to p and add l (Dirichlet 1863/1999, and P. In either case there is a prime greater than pN 10). and so an infinity of primes (Hardy 1938, p. 125). From lemma L2, Dirichlet then concludes a contradiction to Despite this dramatic change in the later editions of his the conclusion that P + 1 is composite. A comparison with Course of Pure Mathematics and his return to Euclid’s Euclid is—once again—difficult because of the assumption original proof, Hardy continued to misunderstand the latter of the completeness of S in Dirichlet. However, one can say as one by contradiction. Hardy gave Euclid’s proof a that, in a sense, Dirichlet is more faithful than Hardy to prominent place in his famous Apology of 1940, where he Euclid’s original proof, because—like Euclid—he first presented basically the same newly corrected version of the (implicitly) considers P + 1 to be prime, but then discards proof from his Course: this possibility under the assumption of A, because P + 1is The proof is by reductio ad absurdum, and reductio ad greater than p. Dirichlet thus even uses expressly lemma L1, absurdum, which Euclid loved so much, is one of a which is hidden in Euclid. That lemma implies that because mathematician’s finest weapons. It is a far finer gambit of the multiplication that produces P + 1, the latter number than any chess gambit: a chess player may offer the must lie outside the original set S. sacrifice of a pawn or even a piece, but a mathematician Symbolically one could write Dirichlet’s indirect proof offers the game (Hardy 1940: 92). MIP-D1863 in the following way: But at the same time and in the same book of 1940, Hardy

L1 L2 explained that Euclid’s theorem is just a fundamental result ðD 1863Þ½A ƒƒƒ!ðP þ 1Þcomposite^½A ƒƒƒ!ðP þ 1Þ about the infinitude of primes and that it was limited in its :composite!:A possible applications. H&W rightly find that presentations such as the one by Dirichlet’s proof seems to me one of the shortest and most Hardy in 1908, which insinuate a proof by contradiction elegant proofs of MCL, that is, the infinitude of primes. It and neglect construction, lack ‘‘simplicity,’’ and that uses—as any other proof that connects to Euclid—the main Euclid’s proof was ‘‘simpler.’’ I believe that Euclid’s proof is, constructive idea, namely to construct and investigate the indeed, ‘‘simpler’’ because it is better pedagogically and number P + 1. However Dirichlet—like Hardy after him— does not resort to wording that makes the notion of ‘‘pri- is more interested in the fact of infinitude (MCL) than in the mality’’ contingent on absurd assumptions. However, if algorithmic problem of finding an additional prime in they mean that Euclid’s proof is simpler because (!) it is finitely many steps. Given that Dirichlet himself had proven constructive, I disagree. On the contrary, one could argue that a proof purely by contradiction is simpler because it much sharper results on the location of the infinitely many 28 primes27 than can be concluded from ECO, this lack of has less content. interest is not surprising. Why is it that Dirichlet’s indirect proof appears more natural and less logically confusing for beginners than Least Common Multiple, Primorials, Factorials: Hardy’s of 1908? One reason is that the (absurd) conclusion The Logical Equivalence of the Proofs that P + 1 should be composite is refuted by a much Let’s return to the question I asked at the beginning: is the simpler numerical counterexample (2 9 3 + 1 = 7). The ‘‘spirit’’ and ‘‘content’’ of Euclid’s proof preserved in mod- stronger pedagogic appeal in Dirichlet seems to me also a ern presentations that use primorials or even factorials, result of Dirichlet following closer to Euclid’s well-known although Euclid uses products of any finite number of proof and using—unlike Hardy—the very intuitive lemma given primes? Does this actually affect the presentation of L1. One could even argue that Dirichlet is more pedagogic the ‘‘substance’’ of Euclid’s result? than Euclid, that is, Dirichlet shows the mathematical H&W rightly criticize remarks such as this one by Tobias instruments that he uses more clearly than Euclid. Indeed, Dantzig (1884–1956), the father of the better-known one should not look at modern presentations of Euclid only mathematician George Dantzig: under the perspective of distortion but also under the point In this proof Euclid introduces for the first time in history of view of logical clarification. what we call today factorial numbers (Dantzig, quoted Apparently, in the end Hardy himself recognized his by Hardy/Woodgold 2009: 47). ‘‘pedagogical error.’’ In the seventh edition of 1938 of his This was even less faithful to Euclid than the use of book (in which he also no longer considered 1 to be a primorials. Of course an historian has to reject such wrong prime) Hardy wrote: attributions to original sources when they come as bluntly There are however, as was first shown by Euclid, infinitely and as unnuanced as in Dantzig’s book. A similar remark many primes. Euclid’s proof is as follows. Let 2, 3, 5, … pN applies to Dickson’s book (1919), which the author calls 29 be all the primes up to pN, and let P = (2.3.5… pN) + 1. himself a ‘‘History of Number Theory.’’ Then P is not divisible by any of 2, 3, 5, …, pN . Hence Now we find another use of factorials in an allusion to either P is prime, or P is divisible by a prime p between pN Euclid’s proof in David Hilbert’s talk ‘‘U¨ ber das

27Namely, the existence of an infinitude of primes in arithmetic sequences. 28For example, as described earlier, it is simpler to prove the actual infinitude of all primes, because of the additional indirect argument that leads to MCL, than to construct the potential infinitude of particular sets of primes with the help of ECO. 29Dickson (1919: 413) says mistakenly that he quotes directly from Heiberg’s Greek edition when he alleges that Euclid used primorials. Thanks go to June Barrow- Green for alerting me to Dickson.

2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 93 Unendliche’’ (On the Infinite) in the context of a discussion this or that property’’ has meaning only as a partial of ‘‘finitism.’’ This talk may have influenced later presen- proposition, that is, as part of a proposition that is more tations of Euclid’s theorem.30 Indeed, in the English precisely determined but whose exact content is unes- translation by the remarkable German mathematician and sential for many applications (Hilbert 1925: 377–378). musician Stefan Bauer-Mengelberg,31 we read: Now Hilbert is known not to have been very history-minded, By means of Euclid’s well-known procedure we can, and he was usually not very concerned about accuracy in his completely within the framework of the attitude we have allusions to the history of his discipline. In the example under adopted [i.e., the ‘‘finitist’’ in Hilbert’s sense as described in discussion here, Hilbert does not give any details about his quote that follows, R. S.-S.], prove the theorem that Euclid’s proof either. However, his claim that Euclid reached between p + landp! + l there certainly exists a new prime two different conclusions, namely ‘‘a much weaker asser- number. This proposition itself, moreover, is completely in tion’’ about infinitely many primes and a stronger one conformity with our finitist attitude. For ‘‘there exists’’ here specifying a finite set where the additional primes exist is in serves merely to abbreviate the proposition: good agreement with what I argued earlier. It remains to be seen whether Hilbert provided an Certainly p + lorp+ 2orp+ 3or… or p! + lisa accurate description of the conclusions that could be drawn prime number. from ‘‘Euclid’s well-known procedure,’’ in particular, how But let us go on. Obviously, to say there exists a prime much the use of factorials changed Euclid’s original argu- number that (1) is [ p and (2) is at the same ment. It is of course obvious that p! is divisible by all natural time B p! + 1 would amount to the same thing, and this numbers \ p, in particular by all primes \ p.Ifp! + 1 is not leads us to formulate a proposition that expresses only a prime, it is divisible by a prime according Euclid VII, 31 part of Euclid’s assertion, namely: there exists a prime (theorem T). This prime must be smaller than the usually number that is [ p. So far as content is concerned, this is very big number p!, but bigger than p, because otherwise a much weaker assertion, stating only a part of Euclid’s the classic contradiction from Euclid’s theorem of a division proposition; nevertheless, no matter how harmless the with a remainder \ 1 occurs. Therefore there must be a transition appears to be, there is a leap into the transfi- prime between p + 1 and p!. Thus the main idea is indeed nite when this partial proposition, taken out of the analogous to Euclid; Hilbert’s prime number p can be context above, is stated as an independent assertion. considered as the biggest prime number in Euclid’s set S. Hilbert has thus shown, with the same argument as Euclid, How can that be? We have here an existential proposition that there exists a prime number bigger than any given with ‘‘there exists.’’ To be sure, we already had one in prime number p and smaller than an upper limit depending Euclid’s theorem. But the latter, with its ‘‘there exists,’’ was, as on a concrete natural number that can be calculated from p. I have already said, merely another, shorter expression for Whereas Euclid proves that there exists a larger set (!) of ‘‘p + 1orp+ 2orp+ 3or...orp!+ 1 is a prime prime numbers than any finite set of prime numbers, Hil- number,’’ bert proves that there exists a larger prime number than any given prime number. Both Euclid and Hilbert provide just as, instead of saying: This piece of chalk is red or concrete boundaries for the new set or the new number. that piece of chalk is red or … or the piece of chalk over Both proofs are logically equivalent, because any finite set there is red, I say more briefly: Among these pieces of of numbers has a largest element and conversely one can chalk there exists a red one. An assertion of this kind, find (for instance with the well-known sieve of Eratosthe- that in a finite totality ‘‘there exists’’ an object having a nes) to a given p all prime numbers smaller than it and thus certain property, is completely in conformity with our one can find any of the upper limits for an additional prime. finitist attitude. On the other hand, the expression ‘‘p + lorp+ 2orp+ 3or… ad infinitum is a prime The Obsolescence of Euclid’s Proof and Some number’’ Concluding Remarks 32 is, as it were, an infinite logical product, and such a I find three main historical reasons for the misrepresenta- passage to the infinite is no more permitted without tion of Euclid’s proof as indirect. special investigation and perhaps certain precautionary The first reason is historical fashion. Indeed, the refer- measures than the passage from a finite to an infinite ence of modern, logically minded mathematicians (from product in analysis, and initially it has no meaning at all. the second half of the 19th and a good part of the 20th In general, from the finitist point of view an existential centuries) to the old ideal of Euclid’s Elements after several proposition of the form ‘‘There exists a number having ‘‘constructive centuries’’ (Descartes through Euler) is not coincidental. They know that Greek mathematicians used

30However, Mesˇ trovic´ (2012: 9) mentions an article by H. Brocard of 1915 that seems to indicate that Charles Hermite already used factorials for the presentation of Euclid’s proof. 31Stefan Bauer-Mengelberg (1927–1996) was a German-born mathematician who worked at I.B.M. and had a simultaneous career as a conductor. He served for a while as assistant conductor to Leonard Bernstein at the New York Philharmonic. Cf. his obituary in The New York Times, published October 28, 1996, also accessible online. 32Editor Jan van Heijenoort adds here the following footnote: ‘‘It is rather a logical sum or disjunction. In the version published in Grundlagen der Geometrie (1930), ‘logisches Produkt’ is replaced by ‘Oder-Verknu¨ pfung’.’’

94 THE MATHEMATICAL INTELLIGENCER indirect proofs widely and thus they read into Euclid what with stronger analytical methods, that in any case there they want to stress. The modern mathematicians are helped always exists a prime between p and 2p.33 That there can at by the fact that Euclid himself in the Elements downplayed the same time exist additional primes smaller than p and the constructive aspects of the proofs or at least did not between p and 2p is shown for p = 7 and the example mention them in the statements of many theorems. This in 2 9 7 + 1 = 15. Thus Chebysev’s result is much sharper particular is the case for Euclid’s theorem of the infinitude than Euclid’s in this case. of primes, which does not mention in its claim the mainly Hardy and Wright (1938) reflect on the conclusions that constructive nature of the proof. From the 1960s, mathe- can be drawn from Euclid’s proof for p(x), the number of maticians such as D. Knuth and G. Po´lya, and philosophers primes below x. They present ECO very close to Euclid’s such as I. Lakatos, have contributed with their work to a spirit as in Hardy’s corrected version discussed earlier. renaissance of constructive and algorithmic methodologies Then they present, based on the use of primorials, upper in mathematics. This was of course gradually supported by limits for the increase of pn which is the value of the nth the rise of data technology and computer science, which prime, and lower limits for p(x). Hardy’s and Wright’s dis- left its mark in prime number theory as well, the prime cussion shows that the two mathematicians were well numbers finding unexpected applications in coding theory aware of the constructive side of Euclid’s proof, but that and so forth, for example, in the banking business. they deemed its potential to be ‘‘absurdly weak’’34 com- The second historical reason for misinterpreting Euclid’s pared to what was already known at the time about the proof as one by contradiction is the weak form of con- distribution of primes, based for instance on the (logarith- structiveness that does not provide an effective procedure mic) prime number theorem (proven in 1896 by Hadamard to find individual primes: Euclid constructed a finite set of and de la Valle´e Poussin) and other more refined results. numbers within which the existence of an additional prime This provides further confirmation of the reason for G. H. was guaranteed; however this additional prime (or primes) Hardy’s mistaken claim that Euclid’s argument was mainly can only be found by testing the elements of this finite set by contradiction. Euclid’s theorem apparently was to him consecutively—and apparently leaving the mathematician only of a general ‘‘philosophical’’ interest, not a concrete without information about a preferential order—for pri- help for finding prime numbers. mality. David Hilbert referred to this weaker notion of One gains still another perspective on the constructive/ construction, which is exhibited by Euclid’s proof, in the algorithmic side of Euclid’s theorem when looking, for context of his ‘‘finitist’’ approach to the foundations of instance, at the book by Boolos, Burgess, and Jeffrey, mathematics in the 1920s and 1930s. The weak construc- which uses factorials: tiveness is also connected to a partial and ‘‘local’’ use of 7.10 Example (The next prime). Let f (x) = the least y such indirect arguments within the proof. One gets the impres- that x \ y and y is prime. The relation x \ y & y is prime is sion that some modern mathematicians looking at Euclid’s primitive recursive, using Example 7.5. Hence the func- weak construction compare it with some modern proofs by tion f is recursive by the preceding proposition. There is a contradiction, not of Euclid’s theorem, but, for instance, in theorem in Euclid’s Elements that tells us that for any given set theory, which indeed are not constructive in any sense. number x there exists a prime y [ x, from which we know There is in my opinion, however, a third historical reason, that our function f is total. But actually, the proof in Euclid no less important, for many modern authors to insinuate an shows that there is a prime y [ x with y B x! + 1. Since indirect proof of Euclid’s theorem and to downplay its con- the factorial function is primitive recursive, the Corollary structiveness. On this historical reason I will elaborate a bit in 7.8 applies to show that f is actually primitive recursive these concluding remarks. Although the infinitude of primes (Boolos, Burgess, and Jeffrey 1980: 79). is still a basic and important fact in number theory, the insight Thus modern work on logic and computability apparently of modern mathematicians into the distribution of primes has drawn—at least on a theoretical level—some inspira- goes far beyond what is indicated in Euclid’s constructive tion from the constructive part of Euclid’s proof, here in proof, which is therefore mathematically obsolete, the entire Boolos et al, in connection with the theory of primitive theorem being a very trivial one. Indeed, Euclid’s proof, both recursive relations and assuming the use of factorials.35 in its original and in its modern forms, offers little information But Boolos et al do not share the concerns of some about the distribution of primes. According to Euclid’s ori- ‘‘heterodox mathematicians who reject certain principles of ginal proof, additional primes can also be sought below the logic.’’ Referring to Hilbert’s ‘‘finitism,’’ they point to the biggest given prime, as the simple examples 3 9 5 + 1 = 16 power of modern, analytical mathematics, interestingly and 2 9 7 + 1 = 15 show. Euclid’s original argument enough in a number-theoretic context: allows, for example, for any prime p [ 2 to conclude the On the plane of mathematical practice, Hilbert insisted, a existence of an additional prime \ 2p. However, Euclid’s detour through the ‘‘ideal’’ is often the shortest route to a proof does not rule out that this prime should be smaller than ‘‘contentful’’ result. (For example, Chebyshev’s theorem p: it was only shown much later by P. L. Chebysev (1850), that there is a prime between any number and its double

33Hardy and Wright (1938: 13). See the later remark quoted from Boolos et al (1980: 238). The theorem is valid even for arbitrary n, not necessarily prime. 34Cf. a similar remark in Mesˇ trovic´ (2012: 30), who calls it a ‘‘horrible bound.’’ 35Cf. also Graham, Knuth, and Patashnik (1989), as quoted earlier, and the note by Mullin (1963) who constructs with the ‘‘Euclidean idea’’ of the number P + 1a sequence of primes that never generates any prime twice. Mullin refers directly back to E. L. Post’s theory of recursively enumerable sets of positive integers and their decision problems (1944). Sequences of primes have certainly played an increasing role in mathematical logic, for instance in the method of Go¨ delization.

2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 95 was proved not in some ‘‘finitistic,’’ ‘‘constructive,’’ REFERENCES directly computational way, but by an argument involving Barrow-Green, J., 2006: ‘‘Much necessary for all sortes of men’’: applying calculus to functions whose arguments and 450 years of Euclid’s Elements in English, BSHM Bulletin 21, values are imaginary numbers.) (Boolos, Burgess, and 2–25. Jeffrey 1980: 238). Boolos, G., Burgess, J. P., and Jeffrey, R. C., 1980: Computability and As is well known, Hilbert’s version of ‘‘finitism’’ was, in Logic, Cambridge: Cambridge University Press. reality, a defense strategy to save classic, infinitist mathe- Dirichlet, P. G. L., 1863: Lectures on Number Theory (supplements by matics. Also, G. H. Hardy was not willing to renounce the R. Dedekind); [Publication of Dedekind’s posthumous edition of use of modern analysis in number theory when in his A 1863 of Dirichlet’s lectures with later supplements I-IX by Mathematician’s Apology he said about Euclid’s theorem: Dirichlet, translated by J. Stillwell], AMS, 1999. The proof can be rearranged to avoid a reductio, and Dickson, L. E., 1919: History of the Theory of Numbers, vol. 1, logicians of some schools would prefer it should be Divisibility and Primality. Washington: The Carnegie Institution. (Hardy 1940: 94). Euclid, 1908, 1926: The Thirteen Books of The Elements, translated It is not fully clear what Hardy meant by rearrangement. I with introduction and commentary by Sir Thomas Heath, New suspect that he did not have in mind a simple rearrangement York: Dover 1956. of Euclid’s proof but rather a strict and formalized rewriting Euclide d’Alexandrie, 1990–2001: Les E´ le´ ments, traduit du texte de of the proof in the language of constructive mathematics. In Heiberg, Traduction et commentaires par Bernard Vitrac;4 any case, this (like Hilbert’s presentation) says less about what Euclid’s proof was (whether by contradiction, or volumes, Paris : Presses Universitaires de France. construction, or both) than about what Hardy wanted it to Graham, Ronald L., Donald E. Knuth, and Oren Patashnik, 1989: have been, determined by his own research interests (which Concrete Mathematics: A Foundation for Computer Science, were certainly not directed toward constructivism) and by Reading, etc. Boston: Addison-Wesley. the spirit of the time. Hardy, M., 2013: Three Thoughts on ‘‘Prime Simplicity,’’ The To Hilbert, the constructive side of Euclid’s proof was Mathematical Intelligencer 35(1), 2. probably of philosophical rather than mathematical inter- Hardy, M., and Woodgold, C., 2009: Prime Simplicity, Mathematical est, and was not of practical concern. Since Hilbert’s times, Intelligencer 31(4), 44–52 much better proofs and methods have been devised, which Hardy, G. H., 1908: A Course of Pure Mathematics, Cambridge: make it fairly easy to find prime numbers that have almost Cambridge University Press. any characteristic one wants them to have. Many modern Hardy, G. H., 1938: A Course of Pure Mathematics, 7th edition, searches for higher primes use special numbers such as Cambridge: Cambridge University Press. Mersenne numbers, for which there exist strong criteria of Hardy, G. H., 1940: A Mathematician’s Apology, Cambridge: Cam- 36 primality. bridge University Press. The examples of Dirichlet, Hardy, Hilbert, and other Hardy, G. H., and E. M. Wright, 1938: An Introduction to Number leading mathematicians of the 19th and 20th centuries Theory, Oxford, Oxford University Press. reading Euclid’s theorem, corroborate a recurring theme Heijenoort, J. v. (ed.), 1967: From Frege to Go¨ del, Cambridge: Harvard in this article, namely that modern views of the history of University Press. mathematics are often colored by current research Hilbert, D., 1925: U¨ ber das Unendliche, Mathematische Annalen 95 interests. (1926), 172, quoted from the English translation by Stefan Bauer- Mengelberg in Heijenoort (1967), 367–392. ACKNOWLEDGMENTS Knorr, W., 1975: The Evolution of the Euclidean Elements, Dordrecht: I thank June Barrow-Green (Open University, Milton Reidel. Keynes, United Kingdom) for careful reading, useful Knorr, W., 1976: Problems in the Interpretation of Greek Number advice, and help with the English. Donald E. Knuth (Stan- Theory: Euclid and the ‘‘Fundamental Theorem of Arithmetic,’’ ford) provided me with detailed and helpful commentary Studies in Hist. and Philosophy of Science Part A 7 (4), 353–368. on a draft of this article. Thanks for discussions are due to Knuth, D. E., 1981: The Art of Computer Programming, vol. 2, Norbert Schappacher (Strasbourg), who also alerted me to Seminumerical Algorithms, 2nd edition: Reading: Addison-Wesley. Vitrac’s modern French edition of Euclid’s Elements.Iam Mesˇ trovic´ , Romeo, 2012: Euclid’s theorem on the infinitude of primes: grateful to the editor, Marjorie Senechal, who helped in A historical survey of its proofs (300 BC to 2012), finding the right focus for the article. http://arxiv.org/abs/1202.3670v2, 5 June 2012, 66p. Mullins, A. A., 1963: Recursive Function Theory (A Modern Look at a Faculty of Engineering and Science Euclidean Idea), Bulletin AMS 69, 737. University of Agder Nikolic´ , M., 1974: The Relation between Eudoxos’ Theory of Propor- Gimlemoen, Postboks 422 tions and Dedekind’s Theory of Cuts, In: Cohen, R. S., J. J. 4604, Kristiansand S Stachel, and M. W. Wartofsky (eds.), For Dirk Struik: Scientific, Norway Historical and Political Essays in Honor of Dirk J. Struik, e-mail: [email protected] Dordrecht: D. Reidel, 225–243.

36Cf., for example, Ribenboim (1996).

96 THE MATHEMATICAL INTELLIGENCER Ribenboim, P., 1996: The New Book of Prime Number Records, New Vitrac, B., 1994: Euclide d’Alexandrie, 1990–2001, volume 2. York: Springer. Weil, A., 1978: History of Mathematics: Why and How? Proceedings of Unguru, S., 1975: On the Need to Rewrite the History of Greek the International Congress of Mathematicians, Helsinki 1, 227– Mathematics, Archive for the History of Exact Sciences 15, 67–114. 236.

Mathematics Pure or Applied

Martin Zerner reports that a few years ago he was scanning a list of mathematical specialties put out by the Conseil National des Recherches Scientifiques, and noticed that one of the categories was

Applications of Pure Mathematics.

With Zerner, we may furrow our brows. Is it an oxymoron? If not, maybe the CNRS ought to have allowed also a category

Purification of Applied Mathematics

–which, if it means anything, surely means something rather inglorious. Like removing from some discipline the stigma of applicability (as if there were such a stigma). Joseph Keller’s dismissal of all such anxieties is

Pure mathematics is a subfield of applied mathematics.

Chandler Davis

2014 Springer Science+Business Media New York, Volume 36, Number 4, 2014 97